Submission date: 14 July 2014.

Abstract:

We extend the characterization of extremal valued fields given in the paper Azgin, S. - Kuhlmann, F.-V. - Pop, F.: Characterization of extremal valued fields, Proc. Amer. Math. Soc. 140 (2012) to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite $p$-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in the paper Kuhlmann, F.-V.: Elementary properties of power series fields over finite fields, J. Symb. Logic 66 (2001) for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with non-perfect residue fields.

Mathematics Subject Classification: Primary 12J20, Secondary 12J10

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Full text arXiv 1407.3759: pdf, ps.